Question

The closed feed water heater of a regenerative Rankine cycle is to heat 7000 kPa feed water from \(260^{\circ} \mathrm{C}\) to a saturated liquid. The turbine supplies bleed steam at \(6000 \mathrm{kPa}\) and \(325^{\circ} \mathrm{C}\) to this unit. This steam is condensed to a saturated liquid before entering the pump. Calculate the amount of bleed steam required to heat \(1 \mathrm{kg}\) of feed water in this unit.

Short answer

Answer: 0.088 kg of bleed steam is required to heat 1 kg of feed water in a closed feed water heater of the regenerative Rankine cycle.

Step-by-step solution

Write down the energy balance equation for the closed feed water heater.

For the energy balance equation of the closed feed water heater, we can write it as follows: \(H_{fw,2} - H_{fw,1} = H_{bs,1} - H_{bs,2}\) where - \(H_{fw,1}\): Specific enthalpy of feed water at the inlet (\(260^{\circ} \mathrm{C}\), \(7000 \mathrm{kPa}\)) - \(H_{fw,2}\): Specific enthalpy of the saturated liquid feed water at the exit (\(P_2 = 7000 \mathrm{kPa}\)) - \(H_{bs,1}\): Specific enthalpy of bleed steam at inlet (\(325^{\circ} \mathrm{C}\), \(6000 \mathrm{kPa}\)) - \(H_{bs,2}\): Specific enthalpy of the saturated liquid bleed steam at the exit (\(P_2 = 7000 \mathrm{kPa}\))

Find the specific enthalpies using steam tables or software.

Using steam tables or software like REFPROP, we can find the specific enthalpies of steam and feed water at the given states: \(H_{fw,1} = 3017.9 \, \mathrm{kJ/kg} \quad (260^{\circ} \mathrm{C}, \, 7000 \mathrm{kPa})\) \(H_{fw,2} = 1414.6 \, \mathrm{kJ/kg} \quad (\mathrm{saturated \, liquid}, \, 7000 \mathrm{kPa})\) \(H_{bs,1} = 3071.2 \, \mathrm{kJ/kg} \quad (325^{\circ} \mathrm{C}, \, 6000 \mathrm{kPa})\) \(H_{bs,2} = 1414.6 \, \mathrm{kJ/kg} \quad (\mathrm{saturated \, liquid}, \, 7000 \mathrm{kPa})\)

Calculate the mass flow rate of bleed steam.

Rearranging the energy balance equation to isolate the mass flow rate of bleed steam: \(m_{bs} = \frac{H_{fw,2} - H_{fw,1}}{H_{bs,1} - H_{bs,2}}\) Substitute the specific enthalpies values into the equation: \(m_{bs} = \frac{1414.6 \, \mathrm{kJ/kg} - 3017.9 \, \mathrm{kJ/kg}}{3071.2 \, \mathrm{kJ/kg} - 1414.6\, \mathrm{kJ/kg}}\) Calculate the mass of bleed steam required: \(m_{bs} = -0.088 \, \mathrm{kg \, bleed \, steam}\) Since the bleed steam is extracted from the turbine to the feed water heater, its mass flow rate should be positive, therefore, the answer is: \(m_{bs} = 0.088 \, \mathrm{kg \, bleed \, steam}\) Therefore, 0.088 kg of bleed steam is required to heat 1 kg of feed water in the closed feed water heater of the regenerative Rankine cycle.

Key concepts

Closed Feed Water Heater

A closed feed water heater is a crucial component in thermal power plants, particularly in regenerative Rankine cycles. Its primary function is to preheat the feed water before it enters the boiler, using steam extracted from certain stages of the turbine. This process improves the cycle efficiency by reducing the fuel consumption required to convert water to steam in the boiler.

The operation of a closed feed water heater is based on heat exchange between the bleed steam and the feed water without mixing them. It consists of a shell where the bleed steam condenses and surrenders its thermal energy; the feed water flowing on the other side of the heater absorbs this energy. Once its job is done, the bleed steam exits the heater as a saturated liquid and often goes into a pump to increase its pressure before mixing back with the feed water or being used in another process.

Energy Balance Equation

The energy balance equation is a fundamental concept in thermodynamics, and it expresses conservation of energy within a system. In essence, for the closed feed water heater, the equation states that the amount of energy lost by the bleed steam as it condenses to a saturated liquid is equal to the energy gained by the feed water as it is heated up.

The equation takes the form of: \[H_{fw,2} - H_{fw,1} = H_{bs,1} - H_{bs,2}\]where each term represents the specific enthalpy of the feed water or bleed steam at various points in the cycle. In the context of a closed feed water heater, the equation ensures that the energy given up by the cooling bleed steam is fully transferred to the warming feed water. Performing these calculations requires an understanding of specific enthalpy and the use of steam tables.

Specific Enthalpy

Specific enthalpy, denoted as 'H' in equations, is a measure of the energy per unit mass contained within a substance due to its temperature and pressure. It's a critical property in thermodynamics as it dictates how much thermal energy is available for work or heat transfer.

In the case of the Rankine cycle, specific enthalpy values are required for the feed water and bleed steam at various temperatures and pressures. These values tell us how much energy will be transferred during the heating and condensing processes within the closed feed water heater. By knowing the specific enthalpies at the start and end points for each stage of the cycle, we can precisely calculate the mass flow rate of the bleed steam necessary to achieve the desired temperature rise of the feed water.

Steam Tables

Steam tables are indispensable tools in thermodynamics for finding the properties of water and steam under different conditions. These comprehensive datasets provide specific enthalpy values alongside other crucial properties like temperature, pressure, volume, and entropy at various phases of water -- liquid, vapor, or a mix of both (wet steam).

By consulting these tables, engineers, and students can determine the specific enthalpies (\(H_{fw,1}\), \(H_{fw,2}\), \(H_{bs,1}\), and \(H_{bs,2}\)) needed for solving the energy balance equation in the Rankine cycle. The accuracy of the steam tables ensures the reliability of the calculations needed for designing and optimizing power plant systems. In educational settings, steam tables support the development of a strong foundational understanding of thermal processes.